I have this very bad misconception regarding when to divide and when not to divide when counting. I have compiled some of these problems(some from the book of Principle of Techniques in Combinatorics and some from my school’s lecture slides). I usually distinguish very clearly between when to strictly use binomial coeffficients only and when to compute binomial coefficients with factorials(to arrange some items). However, after going through the problems, it seems one can apply the multiplication principle on binomial coefficients as well to mimic the behaviour of arrangement, based on the multiplication principle.
Here are the problems and my interpretation of how whether I need to consider any form of repetition given the context. Answers shown and my own perspective will only from the binomial coefficient perspective to highlight the difference.
Highlighted in green are the restrictions for the corresponding sub problems. Highlighted in yellow is the pool to make our subsets from(which is what the binomial coefficient is counting)
From all my previous experiences, I always believe it all bows down to common sense and phrasing in the question whether they want to consider any form of distinction after the restriction they impose, and what is the object we are actually "choosing" and I cannot accurately discern which is the right way to count quite frequently.
Is there a clearer way of explaining the anomaly of seeing division in the pairing problem or any alternate form of phrasing to make the other questions become a scenario that requires us to consider “division to account for lack of emphasis of order” to give me a wider perspective.